# Thread: Show arc lengths are equal.

1. ## Show arc lengths are equal.

Here's a problem I seen somewhere I thought was kind of fun:

Show that the arc length of sin(x) over the interval $\displaystyle [0,2\pi]$ is equal to the circumference of the ellipse $\displaystyle x^{2}+2y^{2}=2$ ".

The arc length integrals are not easily done by elementary means. At least, I don't think so

2. Originally Posted by galactus
Here's a problem I seen somewhere I thought was kind of fun:

Show that the arc length of sin(x) over the interval $\displaystyle [0,2\pi]$ is equal to the circumference of the ellipse $\displaystyle x^{2}+2y^{2}=2$ ".

The arc length integrals are not easily done by elementary means. At least, I don't think so
There is no exact value of the length of an ellipse, but we can prove the two curves have the same length.

3. Very good curvature. That's how I tackled it. Also, the same way it was done on another forum. That is apparently the best way. I will admit, I scratched my head for a little while before it dawned on me to use parametric arc length. Then , it was obvious. Fell right into place

4. Originally Posted by galactus
That's how I tackled it. Also, the same way it was done on another forum. That is apparently the best way. I will admit, I scratched my head for a little while before it dawned on me to use parametric arc length. Then , it was obvious. Fell right into place
An interesting problem and we enjoyed it. Didnt we?

5. I get the feeling you may think I actually didn't solve the problem, but said I did it that way. Sorry, if it looks that way but I would'nt be that dishonest.

I was thinking maybe someone would show another way besides the parametric arc length method. I won't post anymore problems that way.

6. Originally Posted by galactus
I get the feeling you may think I actually didn't solve the problem, but said I did it that way. Sorry, if it looks that way but I would'nt be that dishonest.
I am sorry you think so. The way I solved the problem is rather normal that many people (including you) can do it that way.

7. I'm sorry. No ruffled feathers. I just didn't want you to think that.